Hello everyone, I'm new to these forums. The long story short is I am trying to finish high school via correspondence, and I need to do two math courses, each containing four units. I am on the fourth unit, the first three gave me very few problems. Anyways, here is the problem:

2cos^2x+cosx=0 Solve for the domain (0, 2pi) to nearest hundredth of a radian.

CAST RULE: since cosx is negative, the angle is in quadrants two and three:

pi-2.09=1.05
pi+2.09=5.23

So we have 1.57, 4.71, 1.05, 5.23
-------------------------------------

However when I check my answer on mathway, it shows the solutions as:

1.57 (correct)
4.71 (correct)
2.09 (?)
4.19 (?)

I don't understand why my calculations for the second zero are wrong. If I use quadrants ONE and FOUR (where cosx should be positive), I get 2.09 and 4.19, which is what mathway shows as the correct answers.

Can someone tell me where I'm going wrong here? The second zero calculates to cosx=(-1/2), and cosx is negative in quadrants two and three, not one and four ... I don't see my error.

October 2nd 2008, 04:18 PM

Soroban

Hello, Jeev!

Welcome aboard!

Quote:

Solve for the domain to nearest hundredth of a radian.

Factor: .

Then: .

October 2nd 2008, 04:36 PM

Jeev

Thanks for giving me the solutions, but I am not understanding how you get the last two.

The CAST rule states that when cosx<0, the angle is in quadrants two and three, which would mean:

(pi) - 2.09
(pi) + 2.09

All I need to understand is, for the second zero, if cosx=(-1/2), why are we calculating the angle in quadrants one and four?

October 2nd 2008, 06:36 PM

Prove It

Quote:

Originally Posted by Jeev

Thanks for giving me the solutions, but I am not understanding how you get the last two.

The CAST rule states that when cosx<0, the angle is in quadrants two and three, which would mean:

(pi) - 2.09
(pi) + 2.09

All I need to understand is, for the second zero, if cosx=(-1/2), why are we calculating the angle in quadrants one and four?

When you put the equation into the calculator, it realised that you've said the angle is negative and calculated the angle in the second quadrant for you.

To do it analytically you need to consider the POSITIVE angle (called the FOCUS angle) and make the adjustments using the CAST (or as I call it, All Students Talk Crap) rule.

Consider

Then your focus angle .

Now, since we are told the angle is negative, we need to consider the 2nd and 3rd quadrants.

So our angles are and . This gives us our angles, and .

Understand now?

October 2nd 2008, 07:12 PM

Jeev

Okay, let me see if I got this.

cosx(2cosx+1)

For 2cosx+1

cosx=(-1/2)

We need to calculate the "focus angle" which is

x=arccos(1/2)
x=1.04 (the question wants radians to the nearest hundredth)

NOW at this point we consider the negative sign, which dictates that the angles are in quadrants II and IV:

(pi)-1.04=2.10
(pi)+1.04=4.18

Is this the right way to go about this?

October 2nd 2008, 07:33 PM

Prove It

Quote:

Originally Posted by Jeev

Okay, let me see if I got this.

cosx(2cosx+1)

For 2cosx+1

cosx=(-1/2)

We need to calculate the "focus angle" which is

x=arccos(1/2)
x=1.04 (the question wants radians to the nearest hundredth)

NOW at this point we consider the negative sign, which dictates that the angles are in quadrants II and IV:

(pi)-1.04=2.10
(pi)+1.04=4.18

Is this the right way to go about this?

Yes, but don't do any rounding until the very last step. Otherwise you get roundoff errors. You said the answers were 2.09 and 4.19 after all...